Graham Smith's Civil War Weapons states this on the Gatling on page 236:

In practice the mechanism was unwieldy and unreliable, and the Gatling wasn't adopted by the government, and those that saw wartime service were private purchases.

Burke Davis devoted an entire chapter to the development of a practical machine gun during the was in The Civil War: Strange and Fascinating Facts. The Gatling is discussed on the last page of the chapter (chapters in the book are short; the chapter on the development is 5 pages with the fifth being an illustration). On the use of the Gatling during the war Davis states on page 60:

Gatling tried to interest Lincoln, who by then had moved on to other weapons, and few of the improved guns got into service. General Ben Butler ordered a dozen, and one of these helped kill Confederates near Petersburg as the end of the war drew nigh. Three of the weapons helped guard the New York Times building in the draft riots of July, 1863.

My next place to turn on the subject was the Encyclopedia of the American Civil War edited by David S. Heidler and Jeanne T. Heidler which has an article on the Gatling on pages 817 and 818. The article mentions that in late 1862 Dr. Gatling was reported by Federal agents to have joined the Knights of the Golden Circle, leading to fears he was going to try selling it to the Confederates. He was proven innocent of this charge but it could have possibly harmed his chances of selling the gun to the Federal government. The article also discusses Ordnance Department chief, Colonel James W. Ripley, and his policy of relying on what the Federal government already had rather than purchasing new models produced by inventors (he retired in September 1863 which could have paved the way for weapons like the Gatling). Ripley refused to purchase the weapon stating his reasons were the weapons meticulous loading and cleaning procedures. However price might have been more the cause. The article goes on to say that Butler, after seeing a demonstration of the gun, purchased a dozen Gatling along with 12,000 rounds for $12,000. So that's $1,000 for one Gatling and a thousand rounds for it, a huge sum of money in the 1860s. Eyewitness accounts state that Butler did use the Gatling during his retreat to Bermuda Hundreds in May 1864. This was following the Battle of Drewry's Bluff.

The Navy was a different story from the Army as they did agree to a demonstration of the gun. This would lead Admiral Dahlgren to allow navy commanders to order the Gatling and one did see service as part of Admiral David D. Porter's Mississippi squadron. However this gun never was used in battle. Also few Gatling saw naval service as Dr. Gatling failed to produce the gun in significant numbers.The Library of Congress Civil War Desk Reference, edited by Margaret E. Wagner, Gary W. Gallagher, and Paul Finkelman says on page 488 that both the Gatling Gun and the Coffee Mill gun were extremely rare with neither government purchasing either and thus neither government issued them to the troops in the field. They were purchased privately by officers willing to use their own funds who

were willing to try new equipment and new tactics.

On 516 the book says that the Gatling saw action during the Siege of Petersburg.
The New York Times "On This Day: August 1, 1863" (https://www.nytimes.com/learning/general/onthisday/harp/0801.html) backs up what Davis said about the Gatling being used during the New York Draft Riots. It also reveals that it was the own of the Times that used the Gatlings during the riot